A328598 Number of compositions of n with no part circularly followed by a divisor.

1, 0, 0, 0, 0, 2, 0, 4, 2, 7, 12, 11, 22, 26, 55, 63, 99, 149, 215, 324, 458, 699, 1006, 1492, 2185, 3202, 4734, 6928, 10242, 14951, 22023, 32365, 47557, 69905, 102633, 150983, 221712, 325918, 478841, 703647, 1034103, 1519431, 2233061, 3281003, 4821790, 7085358

Offset: 0

Author

Gus Wiseman, Oct 24 2019

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

Circularity means the last part is followed by the first.

Examples

			The a(5) = 2 through a(12) = 22 compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)
  (3,2)  (3,4)  (5,3)  (4,5)    (4,6)      (3,8)    (7,5)
         (4,3)         (5,4)    (6,4)      (4,7)    (2,3,7)
         (5,2)         (7,2)    (7,3)      (5,6)    (2,7,3)
                       (2,4,3)  (2,3,5)    (6,5)    (3,2,7)
                       (3,2,4)  (2,5,3)    (7,4)    (3,4,5)
                       (4,3,2)  (3,2,5)    (8,3)    (3,5,4)
                                (3,5,2)    (9,2)    (3,7,2)
                                (5,2,3)    (2,4,5)  (4,3,5)
                                (5,3,2)    (4,5,2)  (4,5,3)
                                (2,3,2,3)  (5,2,4)  (5,3,4)
                                (3,2,3,2)           (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The necklace version is A328600, or A318729 without singletons.

The version with singletons is A318726.

The non-circular version is A328460.

Also forbidding parts circularly followed by a multiple gives A328599.

Partitions with no part followed by a divisor are A328171.

Cf. A000740, A008965, A167606, A178470, A318748, A328187, A328508, A328593, A328597, A328601, A328603, A328609.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,0,10}]

PARI

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i,j)->i%j<>0)))} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n > 0) = A318726(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019